Which of the following is//are conservative force (s) ?

Options

A. `overset(to)F=2r^(3) overset(to)r`
B. `overset(to)F=(3(xoverset(^)(i) + y overset(^)j))/((x^(2) +y^(2))^(3//2))`
C. `overset(to)F=(3(yoverset(^)(i) + x overset(^)(j)))/((x^(2) +y^(2))^(3//2)`
D. none of these

Answer

A, B

Explanation

Since, `W=int overset(to)(F) . D overset(to)( r)`
Clearly for forces (a), the integration does not require any information of the path taken.
For (b) `W_( C) = int(3 (x overset(^) (i) + y overset(^)(j) ))/((x^(2) + y^(2))^(3//2)). (dx overset(^)(i) + dy overset(^)(j) ) =3 int (x dx + y dy)/( (x^(2) + y^(2) )^(3//2))`
Taking `x^(2) +y^(2) =t`, we get `2x dx + 2y dy = dt`
`rArr xdx +ydy = (d t)/(2)`
`therefore W_( C)= 3 int(dt//2)/(t^(3//2))= (3)/(2) int (dt)/( t^(3//2))`
which will be independent of path after solving. Hence, options (a) and (b) are conservative forces.